What type of train would we take, would we have to change trains at some point?
You would have to change trains in La Spezia (smallish, easy station), and most likely Pisa (bigger but also easy to get around) looking at the options. you can look up all the options at www.trenitalia.com just put in Vernazza as the starting point and Florence as the arrival point. the options are a combination of Regionale, Intercity IC, and Eurostar. All of them are perfectly comfortable in Italy, even in 2nd class.
Here is the site for Trenitalia http://www.trenitalia.com/ I found this very helpful in planning train travel. Use Vernazza and Fireze as starting and arrival. CHoose the date and hour you want to leave. It looks like most will involve a change in Spezia and Pisa.
You don't have much choice on the type of trains. You will definitely be on a local Regionale from Vernazza to La Spezia Centrale. From La Spezia, you will most likely be on another Regionale train to Pisa where you will most likely change to yet another Regionale train to Florence. It's not an expensive journey. Here's a sample train schedule link: http://orario.trenitalia.com/b2c/nppPriceTravelSolutions.do?car=0&stazin=vernazza&stazout=florence&datag=30&datam=03&dataa=2011&timsh=06&stazin_r=Staz_DA&stazout_r=Staz_A&timsm=06&timsm_r=06&lang=en&nreq=5&channel=tcom&npag=1&lang_r=en&nreq_r=5&channel_r=tcom&npag_r=1&x=36&y=8